Finite Group Databasebeta

← C2×#20275511 #27387674

Subgroups — C2×#20275511 #27387674

Order 13568.

2094 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D53C211yesmaximal, characteristic
2C22×#13133146C211yesmaximal, characteristic, complemented
3C22×C22⋊#88322C211yesmaximal, characteristic, complemented
4C22×#13133129C211yesmaximal, characteristic, complemented
5C22×#13133129C211yesmaximal, characteristic, complemented
6C22×#13133127C211yesmaximal, characteristic, complemented
7#20275511C218yesmaximal, direct product factor
8C22×C22⋊C8×C53C211yesmaximal, characteristic, complemented
9C23×C22⋊D53C411yescharacteristic
10C23×C4×D53C2211yescharacteristic
11C23×#88322C2211yescharacteristic
12C23×#88322C2211yescharacteristic
13C2×C4×C22⋊D53—24no—
14C2×C4×C22⋊D53—24no—
15C2×C4×C22⋊D53C414yes—
16C2×C4×C22⋊D53C2214yes—
17C22×#11719919C411yescharacteristic
18C22×C22⋊Dic53C411yescharacteristic
19C22×#11719921C411yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21C2×#13133146—24no—
22C2×#13133146C2214yescomplemented
23C2×C22⋊#88322—24no—
24C2×C22⋊#88322C2214yescomplemented
25C2×#13133129—24no—
26C2×#13133129—24no—
27C2×#13133129C2214yescomplemented
28C2×#13133129C2214yescomplemented
29C2×#13133127—24no—
30C2×#13133127C2214yescomplemented
31C24×C4×C53C2211yescharacteristic
32C23×C8×C53C2211yescharacteristic
33C23×C8×C53C2211yescharacteristic
34C2×C22⋊C8×C53—24no—
35C2×C22⋊C8×C53C2214yescomplemented
36C24×D53C2×C411yescharacteristic
37C22×C22⋊D53—24no—
38C22×C22⋊D53—24no—
39C22×C22⋊D53—24no—
40C22×C22⋊D53C2×C414yes—
41C23×Dic53C2×C411yescharacteristic
42C23×Dic53C2×C411yescharacteristic
43C23×Dic53C2×C411yescharacteristic
44C22×C4×D53—22no—
45C22×C4×D53—22no—
46C22×C4×D53—22no—
47C22×C4×D53—22no—
48C22×C4×D53—22no—
49C22×C4×D53—22no—
50C22×C4×D53C2×C412yes—
51C22×C4×D53C2312yes—
52C22×#88322—22no—
53C22×#88322—22no—
54C22×#88322—22no—
55C22×#88322—22no—
56C22×#88322—22no—
57C22×#88322D414yescomplemented
58C22×#88322C2312yes—
59C22×#88322C2312yes—
60C4×C22⋊D53—216no—
61C4×C22⋊D53—28no—
62C4×C22⋊D53—28no—
63C2×#11719919—24no—
64C2×#11719919C2×C414yes—
65C2×C22⋊Dic53—24no—
66C2×C22⋊Dic53C2×C414yes—
67C2×#11719921—24no—
68C2×#11719921C2×C414yes—
69C2×C4×Dic53C2×C412yes—
70C2×C4×Dic53D414yes—
71C2×C4×Dic53D414yes—
72C2×C4×Dic53C2312yes—
73#13133146—28no—
74C22⋊#88322—28no—
75#13133129—28no—
76#13133129—28no—
77#13133127—28no—
78C25×C53C2×C411yescharacteristic
79C23×C4×C53—22no—
80C23×C4×C53—22no—
81C23×C4×C53—22no—
82C23×C4×C53—22no—
83C23×C4×C53—22no—
84C23×C4×C53—22no—
85C23×C4×C53C2×C411yescharacteristic
86C23×C4×C53C2×C412yes—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C22×C8×C53—22no—
90C22×C8×C53—22no—
91C22×C8×C53—22no—
92C22×C8×C53—22no—
93C22×C8×C53—22no—
94C22×C8×C53D414yescomplemented
95C22×C8×C53C2312yes—
96C22×C8×C53C2312yes—
97C22⋊C8×C53—28no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53M1614yes—
105C24×C53C22×C412yes—
106C24×C53C22×C411yescharacteristic
107C23×D53—22no—
108C23×D53—22no—
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—22no—
113C23×D53M1614yes—
114C23×D53C22×C412yes—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—24no—
119C22×C4×C53—24no—
120C22×C4×C53—24no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—44no—
128C22×C4×C53—44no—
129C22×C4×C53—44no—
130C22×C4×C53—44no—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53#4112yes—
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C2×C22⋊D53—232no—
141C2×C22⋊D53—28no—
142C2×C22⋊D53—28no—
143C2×C22⋊D53—28no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53C22⋊C414yes—
152C22×Dic53C22⋊C414yes—
153C22×Dic53C22×C412yes—
154C22×Dic53C22×C412yes—
155C22×Dic53C22×C412yes—
156C2×C4×D53—24no—
157C2×C4×D53—24no—
158C2×C4×D53—24no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—22no—
164C2×C4×D53—22no—
165C2×C4×D53—44no—
166C2×C4×D53—44no—
167C2×C4×D53—44no—
168C2×C4×D53—44no—
169C2×C8×C53—28no—
170C2×C8×C53—22no—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—44no—
176C2×C8×C53C2×D414yescomplemented
177C22⋊Dic53—28no—
178#11719921—28no—
179#11719919—28no—
180C4×Dic53C22⋊C418yes—
181C4×Dic53C2×D414yes—
182C4×Dic53C2×D414yes—
183C2×#88322—28no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—44no—
190C2×#88322C2×D414yescomplemented
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—28no—
198C23×C53—28no—
199C23×C53—28no—
200C23×C53—44no—
201C23×C53—44no—
202C23×C53—44no—
203C23×C53—44no—
204C23×C53C2×C22⋊C412yes—
205C23×C53C4×D412yes—
206C23×C53C4×D412yes—
207C23×C53C2×M1612yes—
208C23×C53C2×M1614yes—
209C23×C53C2×M1612yes—
210C23×C53#13112yes—
211C23×C53#13112yes—
212C23×C53C23×C411yescharacteristic
213C22×D53—22no—
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—28no—
220C22×D53—28no—
221C22×D53—28no—
222C22×D53—44no—
223C22×D53—44no—
224C22×D53—44no—
225C22×D53—44no—
226C22×D53C2×M1614yes—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—44no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—48no—
236C2×C4×C53C2×C22⋊C412yes—
237C2×C4×C53C4×D412yes—
238C2×C4×C53C4×D412yes—
239C2×C4×C53#12114yes—
240C2×C4×C53#12114yes—
241C2×C4×C53#12414yes—
242C2×C4×C53#12714yes—
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C22⋊D53—232no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—28no—
255C2×Dic53—28no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—44no—
260C2×Dic53C2×C22⋊C414yes—
261C2×Dic53C2×C22⋊C414yes—
262C4×D53—42no—
263C4×D53—42no—
264C4×D53—42no—
265C4×D53—42no—
266C4×D53—48no—
267C8×C53—24no—
268C8×C53—42no—
269#88322—24no—
270#88322—42no—
271C2×#1955—531nomaximal, Sylow
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—24no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—416no—
279C22×C53—42no—
280C22×C53#39014yes—
281C22×C53#40714yes—
282C22×C53#40814yes—
283C22×C53#43514yes—
284C22×C53#43514yes—
285C22×C53#43514yes—
286C22×C53#43514yes—
287C22×C53C22×C22⋊C411yescharacteristic
288C22×C53C2×C4×D411yescharacteristic
289C22×C53C2×C4×D411yescharacteristic
290C22×C53C22×M1611yescharacteristic
291C22×C53C22×M1611yescharacteristic
292C22×C53C2×#13111yescharacteristic
293C22×C53C2×#13111yescharacteristic
294C2×D53—24no—
295C2×D53—24no—
296C2×D53—24no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—416no—
301C2×D53—42no—
302C4×C53—42no—
303C4×C53#39014yes—
304C4×C53C2×#12111yescharacteristic
305C4×C53C2×#12111yescharacteristic
306C4×C53C2×#12411yescharacteristic
307C4×C53C2×#12711yescharacteristic
308Dic53—24no—
309Dic53—24no—
310Dic53—42no—
311C22×C4×D4—531no—
312C22×C4⋊C8—531no—
313C22×C22⋊C8—531no—
314C22×C22⋊C8—531no—
315C22×C22⋊C8—531no—
316C22×C22⋊C8D5311yescharacteristic, complemented
317C22×#97—531no—
318#1955—538no—
319C2×C53—44no—
320C2×C53#195518yescomplemented
321C2×C53C2×#39011yescharacteristic
322C2×C53C2×#40711yescharacteristic
323C2×C53C2×#40811yescharacteristic
324C2×C53C2×#43511yescharacteristic
325C2×C53C2×#43511yescharacteristic
326C2×C53C2×#43511yescharacteristic
327C2×C53C2×#43511yescharacteristic
328D53—44no—
329C23×D4—531no—
330C24×C4—531no—
331C24×C4C2×D5311yescharacteristic
332C23×C8—531no—
333C23×C8—531no—
334C23×C8C2×D5311yescharacteristic
335C23×C8C2×D5311yescharacteristic
336C2×C4×D4—534no—
337C2×C4×D4—534no—
338C2×C4×D4—1064no—
339C2×C4×D4—1064no—
340C22×C4⋊C4—531no—
341C22×C22⋊C4—531no—
342C22×C22⋊C4—531no—
343C22×C42—531no—
344C2×C4⋊C8—534no—
345C2×C4⋊C8—1064no—
346C2×C22⋊C8—24no—
347C2×C22⋊C8—534no—
348C2×C22⋊C8—534no—
349C2×C22⋊C8—534no—
350C2×C22⋊C8—1064no—
351C2×C22⋊C8—1064no—
352C2×C22⋊C8—1064no—
353C2×C22⋊C8C2×D5314yescomplemented
354C2×#97—534no—
355C2×#97—1064no—
356C53C2×#195511yescharacteristic, complemented, Sylow
357C25—531no—
358C25C4×D5311yescharacteristic
359C22×D4—534no—
360C22×D4—1064no—
361C22×D4—1064no—
362C22×D4—1064no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—532no—
373C23×C4—532no—
374C23×C4—1062no—
375C23×C4—1062no—
376C23×C4—1062no—
377C23×C4—1062no—
378C23×C4—1062no—
379C23×C4—1062no—
380C23×C4C4×D5311yescharacteristic
381C23×C4C4×D5312yes—
382C23×C4C22×D5312yes—
383C23×C4C22×D5311yescharacteristic
384C22×C8—22no—
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—22no—
388C22×C8—22no—
389C22×C8—534no—
390C22×C8—532no—
391C22×C8—532no—
392C22×C8—1062no—
393C22×C8—1062no—
394C22×C8—1062no—
395C22×C8—1062no—
396C22×C8—1062no—
397C22×C8C22⋊D5314yescomplemented
398C22×C8C22×D5312yes—
399C22×C8C22×D5312yes—
400C4×D4—10616no—
401C4×D4—1068no—
402C4×D4—1068no—
403C2×C4⋊C4—534no—
404C2×C4⋊C4—1064no—
405C2×C22⋊C4—534no—
406C2×C22⋊C4—534no—
407C2×C22⋊C4—1064no—
408C2×C22⋊C4—1064no—
409C2×C42—532no—
410C2×C42—534no—
411C2×C42—534no—
412C2×C42—532no—
413C4⋊C8—1068no—
414C22⋊C8—28no—
415C22⋊C8—1068no—
416C22⋊C8—1068no—
417C22⋊C8—1068no—
418#97—1068no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—534no—
426C24—532no—
427C24—1062no—
428C24—1062no—
429C24—1062no—
430C24—1062no—
431C24—1062no—
432C24—1062no—
433C24#1171991214yes—
434C24C2×C4×D5312yes—
435C24C2×C4×D5311yescharacteristic
436C2×D4—1068no—
437C2×D4—1068no—
438C2×D4—10632no—
439C2×D4—1068no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—534no—
457C22×C4—534no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1064no—
469C22×C4—1064no—
470C22×C4—1064no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—2124no—
478C22×C4—2124no—
479C22×C4—2124no—
480C22×C4—2124no—
481C22×C4C2×C4×D5312yes—
482C22×C4C2×C4×D5311yescharacteristic
483C22×C4#1171994512yes—
484C22×C4D4×D5312yes—
485C22×C4D4×D5312yes—
486C22×C4D4×D5312yes—
487C22×C4#1171994712yes—
488C22×C4C2×C22⋊D5312yes—
489C22×C4C23×D5311yescharacteristic
490C2×C8—28no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—44no—
497C2×C8—534no—
498C2×C8—1068no—
499C2×C8—1062no—
500C2×C8—1062no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—2124no—
505C2×C8C2×C22⋊D5314yescomplemented
506C4⋊C4—1068no—
507C22⋊C4—1068no—
508C22⋊C4—1068no—
509C42—538no—
510C42—534no—
511C42—534no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—28no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—28no—
520C23—28no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—534no—
526C23—1062no—
527C23—1062no—
528C23—1062no—
529C23—1068no—
530C23—1062no—
531C23—1062no—
532C23—1062no—
533C23—1068no—
534C23—1068no—
535C23—2124no—
536C23—2124no—
537C23—2124no—
538C23—2124no—
539C23C22⋊C4×D5312yes—
540C23#1313317712yes—
541C23C4×C22⋊D5312yes—
542C23M16×D5312yes—
543C23C2×#1171991214yes—
544C23C2×#1171991212yes—
545C23#1313322312yes—
546C23#1313322512yes—
547C23C22×C4×D5311yescharacteristic
548D4—10632no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—534no—
559C2×C4—534no—
560C2×C4—1068no—
561C2×C4—1068no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—2124no—
570C2×C4—2128no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4C22⋊C4×D5312yes—
576C2×C4#1313317712yes—
577C2×C4C4×C22⋊D5312yes—
578C2×C4#1313318914yes—
579C2×C4#1313319014yes—
580C2×C4#1313320314yes—
581C2×C4#1313321314yes—
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C2×D4×D5311yescharacteristic
584C2×C4C2×D4×D5311yescharacteristic
585C2×C4C22×C22⋊D5311yescharacteristic
586C2×C4C2×#1171994711yescharacteristic
587C2×C4C2×#1171994511yescharacteristic
588C8—24no—
589C8—42no—
590C8—1064no—
591C8—2122no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—416no—
599C22—42no—
600C22—1064no—
601C22—1064no—
602C22—1064no—
603C22—2122no—
604C22—2122no—
605C22—21216no—
606C22—2122no—
607C22—2122no—
608C22#1681052314yes—
609C22#1681058514yes—
610C22#1681058814yes—
611C22#1681069114yes—
612C22#1681069214yes—
613C22#1681069414yes—
614C22#1681069714yes—
615C22C2×C22⋊C4×D5311yescharacteristic
616C22C2×#1313317711yescharacteristic
617C22C2×C4×C22⋊D5311yescharacteristic
618C22C2×M16×D5311yescharacteristic
619C22C22×#1171991211yescharacteristic
620C22C2×#1313322311yescharacteristic
621C22C2×#1313322511yescharacteristic
622C4—42no—
623C4—1064no—
624C4—1064no—
625C4—2122no—
626C4#1681052314yes—
627C4C2×#1313318911yescharacteristic
628C4C2×#1313319011yescharacteristic
629C4C2×#1313320311yescharacteristic
630C4C2×#1313321311yescharacteristic
631C2—44no—
632C2—2124no—
633C2#2027551118yesdirect product factor
634C2C2×#1681052311yescharacteristic
635C2C2×#1681058511yescharacteristic
636C2C2×#1681058811yescharacteristic
637C2C2×#1681069111yescharacteristic
638C2C2×#1681069211yescharacteristic
639C2C2×#1681069411yescharacteristic
640C2C2×#1681069711yescharacteristic
subgroupG/H
centerC23×C4C22×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×C22⋊C8×C53C2
socleC24×C53C22×C4
prime radicalC23×C22⋊D53C4
cosocleC22×C4×C53C24
hypercenterC22×C22⋊C8D53
hypocenterC53C2×#1955
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23×C4 ◁ C22×C22⋊C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#13133146C2semidirect
3. C22×C22⋊#88322C2semidirect
4. C22×#13133129C2semidirect
5. C22×#13133129C2semidirect
6. C22×#13133127C2semidirect
7. #20275511C2direct (N × H)
8. C22×C22⋊C8×C53C2semidirect
22. C2×#13133146C22semidirect
24. C2×C22⋊#88322C22semidirect
27. C2×#13133129C22semidirect
28. C2×#13133129C22semidirect
30. C2×#13133127C22semidirect
35. C2×C22⋊C8×C53C22semidirect
57. C22×#88322D4semidirect
94. C22×C8×C53D4semidirect
176. C2×C8×C53C2×D4semidirect
190. C2×#88322C2×D4semidirect
316. C22×C22⋊C8D53semidirect
320. C2×C53#1955semidirect
353. C2×C22⋊C8C2×D53semidirect
356. C53C2×#1955semidirect
397. C22×C8C22⋊D53semidirect
505. C2×C8C2×C22⋊D53semidirect
633. C2#20275511direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C4×C22⋊D532
C22×#131331462
C22×C22⋊#883222
C22×#131331292
C22×#131331272
#202755112
C22×C22⋊C8×C532
C2×#195553